Numerical Solution of Generalized Lyapunov Equations. Thilo Penzl. March 13, Abstract

Transcription

1 Numerical Solution of Generalized Lyapunov Equations Thilo Penzl March 3, 996 Abstract Two ecient methods for solving generalized Lyapunov equations and their implementations in FORTRAN 77 are presented. The rst one is a generalization of the Bartels{Stewart method and the second is an extension of Hammarling's method to generalized Lyapunov equations. Our LAPACK based subroutines are implemented in a quite exible way. They can handle the transposed equations and provide scaling to avoid overow in the solution. Moreover, the Bartels{Stewart subroutine oers the optional estimation of the separation and the reciprocal condition number. A brief description of both algorithms is given. The performance of the software is demonstrated by numerical experiments. Key Words: Mathematical software, generalized Lyapunov equation, generalized Stein equation, condition estimation. AMS(MOS) subject classications: 65F5, 65F5, 93B4, 93B5 Basic Properties of Generalized Lyapunov Equations Lyapunov equations play an essential role in control theory. In the past few years its generalizations, the generalized continuous{time Lyapunov equation (GCLE) A T XE + E T XA = Y () and the generalized discrete{time Lyapunov equation (GDLE) or generalized Stein equation A T XA E T XE = Y; (2) have received a lot of interest. A, E, and Y are given real n n matrices. The right hand side Y is symmetric and so is the solution matrix X if the equation has a unique solution. We can consider the GCLE and the GDLE as special cases of the generalized Sylvester equation R T XS + U T XV = Y; (3) where in general, X and Y are n m matrices. Since equation (3) is linear in the entries of X, it can be written as a system of linear equations. To this end, the entries of X are usually arranged in a vector by stacking the columns of X = (x ij ) nm. This is done by the mapping vec, which is dened as vec(x) = (x ; : : : ; x n ; x 2 ; : : : ; x n2 ; : : : ; x m ; : : : ; x nm ) T : It can easily be shown that (3) is equivalent to S T R T + V T U T vec(x) = vec(y ); (4) where denotes the Kronecker product of two matrices, e.g. [3]. The order of this system is nm. Therefore, it is not practicable to nd X by solving the corresponding system of linear equations unless n and m are very small. Technische Universitat Chemnitz{Zwickau, Fakultat fur Mathematik, 97 Chemnitz, FRG, E{mail: tpenzl@mathematik.tu-chemnitz.de. Supported by Deutsche Forschungsgemeinschaft, research grant Me 79/7{ Singulare Steuerungsprobleme.

2 The solvability of (3) depends on the generalized eigenstructure of the matrix pairs (R; U) and (V; S). A matrix pencil R U is called regular i there exists a pair of complex numbers ( ; ) such that R U is nonsingular. If Rx = Ux holds for a vector x 6=, the pair (; ) 6= (; ) is a generalized eigenvalue. Two generalized eigenvalues (; ) and (; ) are considered to be equal i =. The set of all generalized eigenvalues of the pencil R U is designated by (R; U). The following theorem, e.g. [3], gives necessary and sucient conditions for the existence and uniqueness of the solution of (3). Theorem The matrix equation (3) has a unique solution if and only if. R U and V S are regular pencils and 2. (R; U) \ ( V; S) = ;. Let ( i ; i ) denote the generalized eigenvalues of the matrix pencil A E. For simplicity, we switch to the more conventional form of a matrix pencil A E, whose eigenvalues are given by i = i = i with i = when i =. Applying the above theorem to () and (2) gives the following corollary. Corollary Let A E be a regular pencil. Then. the GCLE () has a unique solution if and only if all eigenvalues of A E are nite and i + j 6= for any two eigenvalues i and j of A E, 2. the GDLE (2) has a unique solution if and only if i j 6= for any two eigenvalues i and j of A E (under the convention = ). As a consequence, singularity of one of the matrices A and E implies singularity of the GCLE (). If both A and E are singular, the GDLE (2) is singular too. Thus, for an investigation of the unique solution of a generalized Lyapunov equation we may assume one of the matrices A and E to be nonsingular. Since A and E play a symmetric role (up to a sign in the discrete{time case), we expect E to be invertible. Under this assumption equations () and (2) are equivalent to the Lyapunov equations and (AE ) T X + X(AE ) = E T Y E (5) (AE ) T X(AE ) X = E T Y E ; (6) respectively, and the classical result about the positive denite solution of the stable Lyapunov equation [4] remains valid for the generalized equations. Theorem 2 Let E be nonsingular and Y be positive denite (semidenite).. If Re( i ) < for all eigenvalues i of A E, then the solution matrix X of the GCLE () is positive denite (semidenite). 2. If j i j < for all eigenvalues i of A E, then the solution matrix X of the GDLE (2) is positive denite (semidenite). There may be a denite solution in the discrete{time case even if E is singular. If j i j < for all eigenvalues i of E A, then the solution matrix X is negative denite (semidenite). But this, of course, means nothing but reversing the roles of A and E. Finally, it should be stressed that the reduction of a generalized equation () or (2) to a standard Lyapunov equation (5) or (6), respectively, is mainly of theoretical interest. If E is possibly ill{conditioned, this is not a numerically feasible approach to solve the generalized equation numerically. In the sequel we present two methods which solve generalized Lyapunov equations without inverting E. 2

3 2 Algorithms for Generalized Lyapunov Equations 2. A Generalization of the Bartels{Stewart Method The algorithm described in this section is an extension of the Bartels{Stewart method [2] to generalized Lyapunov equations (see also [3], [6], [2]). We restrict ourself to the GCLE. The derivation of the algorithm for the GDLE is straightforward. First the pencil A E is reduced to real generalized Schur form A s E s by means of orthogonal matrices Q and Z (QZ{algorithm [8]): A s = Q T AZ; E s = Q T EZ such that A s is upper quasitriangular and E s is upper triangular. Dening Y s = Z T Y Z; X s = QXQ T leads to the reduced Lyapunov equation A T s X se s + E T s X sa s = Y s ; () which is equivalent to (). Let A s, E s, Y s, and X s be partitioned into p by p blocks with respect to the subdiagonal structure of the upper quasi-triangular matrix A s. Especially, the main diagonal blocks are or 2 2 matrices according to real eigenvalues or conjugate complex eigenvalue pairs of the pencil A s E s, respectively. A s = Y s = A A p.... O A pp Y Y p..... Y p Y pp C A ; E s B C A ; X s = E E p.... O E pp X X p..... X p E pp Now () can be solved by a block forward substitution, which is more complicated to derive than that utilized in the Bartels{Stewart method for the standard Lyapunov equation. Since X s is symmetric, only p(p + )=2 Sylvester equations A T kkx kl E ll + E T kkx kl A ll = ^Y kl () of order at most 2 2 with right hand side matrices ^Y kl = Y kl + k;l X i=;j= (i;j)6=(k;l) A T ik X ij E jl + E T ik X ija jl need to be solved. According to (4) the solution X ij is found by solving the corresponding system of linear equations E T ll A T kk + A T ll Ekk T vec(xkl ) = vec( ^Ykl ): We determine the blocks in the upper triangle of X s in a row{wise order, i.e., we compute successively X,..., X p, X 22,..., X 2p,..., X pp. Computing the matrices ^Ykl by (2) would result in an overall complexity O(n 4 ). This can be avoided by a somewhat complicated updating technique based upon the following expansion of equation (2) kx X ^Y kl = Y kl A T l X X ij E jl A + E T l X ij A jl AA + i= j= k X A T ik X il E ll + Eik T X ila ll : i= 3 j= C A ; C A : (7) (8) (9) (2)

4 We determine ^Ykl in 2k sweeps Y () kl = Y kl ; kl = Y (2i 2) + A T ik Y (2i ) kl l j= X ij E jl A + E T X l X ij A jl A ; i = ; : : :; k; j= Y (2i) kl = Y (2i ) kl + A T ik X ile ll + E T ik X ila ll ; i = ; : : :; k ; ^Y kl = Y (2k ) kl : Our implementation of the algorithm computes Y (2i ) kl (k i) right before solving () for X il. After X il is known, Y (2i) kl (k > i) can be computed. What decreases the complexity to O(n 3 ) is that Y (2i ) i;l,..., Y (2i ) (the elements below the main diagonal are not of interest) can be updated l;l simultaneously, where we have to compute the terms P l j= X ije jl and P l j= X ija jl only once. Note that the blocks of the strict lower triangle of X s appearing in each of both sums are given by symmetry. Moreover, it is apparent that the updating technique described above can be realized by operating on the array Y without the need for further workspace. Finally, the solution matrix X is obtained from X s by (9). Lyapunov equations whose coecient matrices A and E are replaced by its transposed matrices can be solved in a similar fashion by block backward substitution. 2.2 Estimation of the Separation and the Condition Number The software for the Bartels{Stewart method provides optional estimates for both the separation and the condition number of the Lyapunov operator. Especially the latter is important for estimating the accuracy of the computed solution. In the continuous{time case the separation is dened as sep c = sep c (A; E) = min jjxjj F = A T XE + E T XA F : Due to the orthogonal invariance of the Frobenius norm, this quantity is not altered by the transformations (7) and (8). Therefore, the estimate can be obtained from the reduced equation () which lowers the computational cost signicantly. According to (4), sep c = K ; c 2 if K c = Es T AT s + AT s ET s is invertible. The quantity K c is estimated by an algorithm due to Higham [] based on Hager's method [9]. Actually, this method yields an estimate for the {norm. This is a quite good approximation of K c 2, since it deviates from K c by no more than a factor n. The algorithm, which is available as routine DLACON in LAPACK [], requires the solution of a few (say 4 or 5) generalized Lyapunov equations A T s X s E s + E T s X s A s = Y s or A s X s E T s + E sx s A T s = Y s. Finally, an estimate for the condition number of the generalized Lyapunov operator, which is represented by the corresponding matrix K c, is provided by cond(k c ) 2 ja s j F jje s j F sep c : Again, note that jajj F = jja s jj F and jje j F = ja s j F. The separation of the discrete{time Lyapunov operator is sep d = sep d (A; E) = min A T XA E T XE jjxjj F = F = K d 2 with K d = A T s A T s Es T Es T. After approximating sep d by the reciprocal estimate for K an estimate for the condition number is gained from cond(k d ) jja sjj 2 F + je s jj 2 F sep d : d 4

5 2.3 A Generalization of Hammarling's Method The method due to Hammarling is an alternative to the Bartels{Stewart method in case the Lyapunov equation to be solved is stable and its right hand side is semidenite. In [] Hammarling suggested that his method can be extended to generalized Lyapunov equations. We will present such a generalization in the sequel. We assume the pencil A E to be stable, i.e., its eigenvalues must lie in the open left half plane in the continuous{time case or inside the unit circle in the discrete{time case. If these conditions are met, the solutions X of the GCLE A T XE + E T XA = B T B (3) and the GDLE A T XA E T XE = B T B (4) are positive semidenite (see Theorem 2). In general, B is a real m n matrix, while A, E, and X are real square matrices of order n. The Cholesky factor U of the solution X = U T U can be found without rst computing X and without forming the matrix product B T B. We focus on the continuous{time equation (3) and give a note about the dierences in the discrete{time case. Similar to the Bartels{Stewart method, the algorithm consists of three parts. First the equation is transformed to a reduced form by means of the orthogonal matrices resulting from the generalized Schur factorization. After solving the reduced equation, the solution is retrieved by back transformation. Applying the QZ{algorithm to the pencil A E yields orthogonal matrices Q and Z such that A s = Q T AZ is upper quasitriangular and E s = Q T EZ is upper triangular. This leads to the reduced equation A T s U T s U s E s + E T s U T s U s A s = B T s B s ; (5) where the n n upper triangular matrix B s on the right hand side is formed as follows. If m n, the matrix B s is obtained from the rectangular QR{factorization Bs BZ = Q B ; where Q B is an orthogonal matrix of order m. Otherwise, we partition BZ as BZ = ^B ^B2 ; where ^B is an m m matrix with the QR{factorization ^B = Q B ^B3 : Thus, B s is given by B s = ^B3 Q T B ^B 2 : To solve the reduced equation (5), we partition the involved matrices as A A A s = 2 E E ; E A s = 2 ; 22 E 22 B B B s = 2 U U ; U B s = 2 : 22 U 22 The upper left blocks are p p matrices (p = ; 2), where p = 2 i the pencil A E has a pair of complex conjugate eigenvalues. Provided that U is nonsingular, the above partitioning leads to the following formulas A T U T U E + E T U T U A = B T B ; (6) A T 22 U T 2 + E T 22 U T 2 M = B T 2 M 2 A T 2 U T E T 2 U T M ; (7) A T 22 U T 22 U 22E 22 + E T 22 U T 22 U 22A 22 = B T 22 B 22 B T 2 B 2 (A T U T + 2 AT U T )(U 22 2 E 2 + U 2 E 22 ) (E T 2 U T + E T 22 U 2)(U T A 2 + U 2 A 22 ) = B T 22 B 22 yy T (8) 5

6 with the auxiliary matrices M = U A E (9) M 2 = B E (2) y = B T 2 (E2U T T + E22U T 2)M T T 2 : As mentioned in Section the matrix E is nonsingular so that the inverse of E exists. Moreover, it can be proved that U is nonsingular as well if B 6=. The above equations enable the entries of U s to be determined recursively. The block U is gained from the stable Lyapunov equation (6). Afterwards, the matrices M and M 2 are determined. Solving (6) for the case p = 2 is described later in an extra paragraph. After computing U, M, and M 2, the generalized Sylvester equation (7) is solved for U2. T If U is nonsingular, the existence of a unique solution U T 2 is guaranteed by Theorem. It remains to nd the Cholesky factor B22 ~ of the right hand side matrix of equation (8) B T 22B 22 + yy T = ~ B T 22 ~ B 22 : The upper triangular matrix B ~ 22 is cheaply obtained from the QR{factorization of B22 ~B22 y T = Q ~B ; where Q ~B is an orthogonal matrix and is the zero matrix which consists of p rows. The resulting equation A T 22U T 22U 22 E 22 + E T 22U T 22U 22 A 22 = B T 22B 22 has the same structure as (5) but its size is lowered by p. Hence, all entries of U s can be determined recursively. After the reduced equation (5) is solved, we nd the upper triangular solution matrix U of equation (3) by the QR{factorization of U s Q T = Q U U with the orthogonal matrix Q U. Of course, the latter need not be accumulated. The next part of this section is addressed to the problems caused by the non{real eigenvalues of the pencil A E when p = 2. The procedure for solving the 2 2 Lyapunov equation (6) is similar to that described above for the equation of order n. For an explanation of the algorithm we make use of complex computation. Nevertheless, in our implementation complex operations are emulated by real ones. First the pencil A E is reduced to Schur form by means of unitary matrices ^Q and ^Z such that ^Q H a a A ^Z = ^A = 2 ; a 22 ^Q H e e E ^Z = ^E = 2 : e 22 Let B ^Z = ^QB ^B be the QR{factorization of B ^Z with an unitary matrix ^QB, for which the entries on the main diagonal of the upper triangular matrix ^B are real and non{negative. Now the reduced form of (6) can be written as ^A H ^U H ^U ^E + ^EH ^U H ^U ^A = ^BH ^B : After partitioning ^B and ^U as follows b b ^B = 2 u u ; ^U = 2 ; b 22 u 22 we nd the entries of ^U from = p a e e a ; u = b ; 6

7 u 2 = b 2 + (a e 2 + a 2 e )u a 22 e + e 22 a ; 2 = p a 22 e 22 e 22 a 22 ; y = b 2 e (e 2 u + e 22 u 2 ); u 22 = 2 qb jyj2 : Finally, the upper triangular solution U of (6) is obtained by the QR{factorization ^U ^Q = QU U ; where Q U is a unitary matrix. It should be stressed that if p = 2 care is needed when computing the auxiliary matrices M and M 2 since U may be ill conditioned. This is the case when the pair (E T AT ; E T BT ) is near to an uncontrollable one. For a further discussion of this issue we refer to Section 6 in []. In our implementation we followed the procedure proposed there. For the discrete{time equation (4) most of the algorithm is similar to that for the continuous{ time equation. An essential dierence is the solution of the reduced equation A T s U T s U s A s E T s U T s U s E s = B T s B s : (2) Here the recursion is based on the formulas A T U T U A E T U T U E = B T B (22) A T 22U T 2M E T 22U T 2 = B T 2M 2 + E T 2U T A T 2U T M A T 22 U T 22 U 22A 22 E T 22 U T 22 U 22E 22 = B T 22 B 22 yy T ; where M and M 2 are dened as in (9) and (2), respectively. The matrix y is given by y = B T 2 A T 2U T + A T 22U T 2 C; where C is a matrix which fulls M 3 = I 2p M2 M M2 M T = CC T : From (9), (2), and (22) we obtain M T 2 M 2 + M T M = I p which leads to M3 2 = M 3. Hence, the symmetric matrix M 3 is the orthogonal projector onto span((m T 2 M T ) T )?. Thus, the 2p 2p matrix M 3 is actually positive semidenite and has rank p. Consequently, the factor C is a 2p p matrix and the matrix y consists of only p columns. 3 Software Implementation Our routines for solving the generalized Lyapunov equation possess some new features that should be mentioned here. Both Hammarling's method and the Bartels{Stewart method are implemented in a way that enables the transposed equations to be solved without transposing the involved matrices explicitly. Furthermore, our routines can benet from Schur factorizations of the pencil A E, which have been computed prior to calling the routine. To keep storage requirements low, the input right hand side and the output solution share the same array. This, of course, results in the loss of the right hand side matrix. An output parameter for scaling of the solution is provided to guard against overow. For the Bartels{Stewart method the symmetric right hand side matrix Y may be supplied as upper or lower triangle. In any case the full solution matrix is returned. Moreover, an optional estimation of the separation and the reciprocal condition number is provided. Of course, when solving the generalized Lyapunov equation via Hammarling's algorithm the condition estimator in the Bartels{Stewart routine can be utilized to detect ill{conditioning in the equation. The number of ops required by the routines is given by the following table. It strongly depends on whether the generalized Schur factorization of the pencil A E is supplied (FACT =.TRUE.) or not, when calling one of both main routines. The op estimate 66n 3 for the QZ{algorithm, which delivers this factorization, is taken from [8]. We split up the op count for the Bartels{Stewart 7

8 method into the three possible cases, where the solution (JOB='X'), the separation (JOB='S'), or both quantities (JOB='B') are to be computed. Note that we count a single oating point arithmetic operation as one op. The quantity c is an integer number of modest size (say 4 or 5). Method FACT=.TRUE. FACT=.FALSE. JOB='B' (26 + 8c)=3 n 3 ( c)=3 n 3 Bartels{Stewart JOB='S' 8c=3 n 3 (98 + 8c)=3 n 3 Hammarling JOB='X' 26=3 n 3 224=3 n 3 m n (3n 3 + 6mn 2 (2n 3 + 6mn 2 +6m 2 n 2m 3 )=3 +6m 2 n 2m 3 )=3 m > n (n 3 + 2mn 2 )=3 (29n 3 + 2mn 2 )=3 Number of ops required by the routines. Our implementation of the Bartels{Stewart algorithm is backward stable if the eigenvalues of the pencil A E are real. Otherwise, linear systems of order at most 4 are involved into the computation. These systems are solved by Gauss elimination with complete pivoting. The loss of stability in such eliminations is rarely encountered in practice. To our knowledge, there are no backward stability results for Hammarling's method for the standard Lyapunov equation. The source code is implemented in FORTRAN 77 and it meets the programming standards of the Working Group on Software [5]. It makes use of BLAS{routines and routines available in LAPACK 2. []. Although BLAS{routines of level 3 are used the algorithms are basically of level and 2. The type COMPLEX is not used. Complex computation is avoided or emulated by real operations. The remainder of this section gives a brief survey of the subroutine organization. An interface specication of both main routines is enclosed in the appendix. Bartels{Stewart method DGLP Main routine. To be called by the user. DGLPRC Solves the reduced continuous{time equation (). DGLPRD Solves the reduced discrete{time equation. DGELUF LU{factorization of a square matrix with complete pivoting. DGELUS Back substitution for linear systems whose coecient matrix has been factorized by DGELUF. Provides scaling. DMTRA Transposes a matrix. DZTAZ Computes Z T AZ or ZAZ T for a symmetric matrix A. Hammarling's method DGLPHM Main routine. To be called by the user. DGHRC Computes the Cholesky factor of the solution of the reduced continuous{time equation (5). DGHRD Computes the Cholesky factor of the solution of the reduced discrete{time equation (2). DGHNX2 Solves the matrix equation A T XC + E T XD = Y or AXC T + EXD T n m matrix X (m = ; 2). Provides scaling. = Y for the DGH2X2 Hammarling's method for the 2 2 Lyapunov equation in case the matrix pencil has complex conjugate eigenvalues. Provides scaling. DCXGIV Computes parameters for the complex Givens rotation. 8

9 DGELUF LU{factorization of a square matrix with complete pivoting. DGELUS Back substitution for linear systems whose coecient matrix has been factorized by DGELUF. Provides scaling. Note that the subroutines DGELUF and DGELUS are contained in LAPACK 2. under the names DGETC2 and DGESC2, respectively. 4 Numerical Experiments In this section we demonstrate the performance of our software applied to two sample problems. Both depend on a scale parameter t which aects the condition number of the equation. We compare the results obtained by the Bartels{Stewart subroutine DGLP and the Hammarling subroutine DGLPHM with those of the established subroutines SYLGC and SYLGD [7]. The tests were carried out on a HP Apollo series 7 workstation under IEEE double precision and machine precision 2:22 6. Example [7]: The matrices A and E are dened as A = (2 t )I n + diag(; 2; 3; : : :; n) + U T n E = I n + 2 t U n in the continuous{time case and A = 2 t I n + diag(; 2; 3; : : :; n) + U T n E = I n + 2 t U n in the discrete{time case, where I n is the n n identity matrix and U n is an n n matrix with unit entries below the diagonal and all other entries zero. These systems become increasingly ill{conditioned as the parameter t increases. In all cases Y is dened as the n n matrix that gives a true solution matrix X of all unit entries. We generated the above problems for a medium size (n = ) and various values of the parameter t. The following table shows the relative errors jj ^X XjjF = jxjj F, where X is the known true solution and ^X is the computed solution aected by roundo. Since the pencil A E is in general not stable, the Hammarling subroutine is not applied to this example. Cont.{time Eq. Discr.{time Eq. t DGLP SYLGC DGLP SYLGD 7.478E{3 2.34E{2.267E{ E{3 4.42E{ E{2.34E{2.E{ 2.3E{8.94E{9 2.72E{ E{ E{ E{ E{6.5E{6 4.46E{3 4.42E{3 7.63E{3 { Example. n =. Relative error. For t = 4 the subroutine SYLGD returned an error ag. For problems of a smaller size (n = ) we compare the estimates for the separation SEP and the reciprocal condition number RCOND with the 'true' values computed applying the singular value decomposition (SVD) to the corresponding Kronecker product matrix. Note that SYLGC and SYLGD do not return estimates for the separation. Cont.{time Eq. Discr.{time Eq. t DGLP true DGLP true 3.685E{.48E{ 2.492E+.49E+.952E{ E{4 3.99E{ E{4 2.97E{ E{7 3.85E{ E{ E{ E{ 3.725E{9 9.33E{ 4.88E{ E{ E{2 9.95E{3 Example. n =. Estimates for the separation. 9

10 Cont.{time Eq. Discr.{time Eq. t DGLP SYLGC true DGLP SYLGD true.98e{3.83e{3 3.83E{3 4.9E{3 7.4E{3.987E{2.699E{5.597E{ E{ E{ E{6.375E{5 2.66E{8.589E{8 4.44E{8 8.67E{9 2.87E{9.339E{8 3.62E{.55E{ 4.337E{ 8.467E{2 2.75E{2.38E{ 4.582E{4.53E{4 4.23E{ E{ E{5.286E{4 Example. n =. Estimates for the reciprocal condition number. Example 2: This example is generated in a way that enables setting the eigenvalues of the pencil A E, which is advantageous for two reasons. In contrast to the previous example it is guaranteed that the pencil has both real and conjugate complex eigenvalues. On the other hand stable pencils can easily be constructed to include the Hammarling subroutine into the comparison. If n = 3q we choose the n n matrices A and E as A = V n diag(a ; : : : ; A q )W n ; A i = E = V n W n s i t i t i t i t i where V n and W n are n n matrices containing only zeroes and ones. The unit entries of V n are on and below the anti{diagonal, those of W n are on and below the diagonal. The parameters s i and t i determining the eigenvalues of the pencil are chosen as s i = t i = t i in the continuous{time case and s i = =t i, t i = p 2s i =2 in the discrete{time case. The semidenite right hand side is formed as the normal matrix A Y = B T B; B = (; 2; : : :; n): The drawback of this example is that the true solution is not known. Therefore, the accuracy of the computed solution ^X is measured by the relative residual dened as jja T ^XE+E T ^XA+Y jjf = jy j F in the continuous{time case and jja T ^XA E T ^XE+Y jjf = jy jj F in the discrete{time case. But this, of course, is a worse criterion compared to the relative error in case the equation is ill{conditioned. The following results were obtained for problems of size n = 99. Cont.{time Eq. Discr.{time Eq. t DGLP DGLPHM SYLGC DGLP DGLPHM SYLGD E{ E{4 3.68E{4.76E{3.72E{ E{5.2.66E{3.28E{ E{4.85E{.844E{ 4.42E{ E{ E{ 3.96E{ E{ E{9 9.92E E{ 4.47E{ 2.423E{ 3.328E{5.4E{ E{ E{ E{9 9.35E{9 { { { Example 2. n = 99. Relative residual. For t = :8 in the discrete{time case each of the subroutines returned an error ag. Finally, the CPU{times obtained by means of the LAPACK{subroutine SECOND are compared for the parameters n = 99 and t = :2. They are split up into the times required for the three main stages of the computation, the transformation of the equation to the reduced form (T), the solution of the reduced equation (RE), and the back transformation of the solution (BT). Cont.{time Eq. Discr.{time Eq. DGLP DGLPHM SYLGC DGLP DGLPHM SYLGD T RE BT Total Example 2. n = 99. t = :2. CPU{times in seconds.

11 Concerning the accuracy of the computed solutions for both examples the involved subroutines do not dier signicantly. The estimates of the condition number obtained by DGLP and SYLGC/SYLGD are of comparable size as well. A distinct merit of our subroutines are the shorter CPU{times. This is mainly caused by the fact that SYLGC/SYLGD make use of several older LINPACK [4] and EISPACK [5] subroutines. Especially, the QZ{subroutine DGEGS from LAPACK invoked by our subroutines is often faster than the modied EISPACK subroutines QZHESG, QZITG, and QZVALG utilized by SYLGC/SYLGD by a factor 6. 5 Acknowledgements I wish to thank Andras Varga for providing his subroutines DMTRA and DZTAZ and the subroutines DGELUF and DGELUS written by Bo Kagstrom and Peter Poromaa.

12 A Subroutine Description A. Bartels{Stewart Method A.. Purpose To solve for X either the generalized continuous{time Lyapunov equation op(a) T X op(e) + op(e) T X op(a) = scale Y (23) or the generalized discrete{time Lyapunov equation op(a) T X op(a) op(e) T X op(e) = scale Y (24) where op(m) is either M or M T for M = A; E and the right hand side Y is symmetric. A, E, Y, and the solution X are N by N matrices. scale is an output scale factor, set to avoid overow in X. An estimation of the separation and the reciprocal condition number of the Lyapunov operator is provided. A..2 Specication A..3 SUBROUTINE DGLP( JOB, DISCR, FACT, TRANS, N, A, LDA, E, LDE, * UPPER, X, LDX, SCALE, Q, LDQ, Z, LDZ, IWORK, * RWORK, LRWORK, SEP, RCOND, IERR ) Arguments In N { INTEGER. Argument List The order of the matrix A. N. A { DOUBLE PRECISION array of DIMENSION (LDA,N). If FACT =.TRUE., then the leading N by N upper Hessenberg part of this array must contain the generalized Schur factor A s of the matrix A. A s must be an upper quasitriangular matrix. The elements below the upper Hessenberg part of the array A are not referenced. If FACT =.FALSE., then the leading N by N part of this array must contain the matrix A. Note: this array is overwritten if FACT =.FALSE.. LDA { INTEGER. The leading dimension of the array A as declared in the calling program. LDA N. E { DOUBLE PRECISION array of DIMENSION (LDE,N). If FACT =.TRUE., then the leading N by N upper triangular part of this array must contain the generalized Schur factor E s of the matrix E. The elements below the upper triangular part of the array E are not referenced. If FACT =.FALSE., then the leading N by N part of this array must contain the coecient matrix E of the equation. Note: this array is overwritten if FACT =.FALSE.. LDE { INTEGER. The leading dimension of the array E as declared in the calling program. LDE N. X { DOUBLE PRECISION array of DIMENSION (LDX,N). If JOB = 'B' or 'X', then the leading N by N part of this array must contain the right hand side matrix Y of the equation. On entry, either the lower or the upper triangular part of this array is referenced (see mode parameter UPPER). 2

13 If JOB = 'S', X is not referenced. Note: this array is overwritten if JOB = 'B' or 'X'. LDX { INTEGER. The leading dimension of the array X as declared in the calling program. LDX N. Q { DOUBLE PRECISION array of DIMENSION (LDQ,N). If FACT =.TRUE., then the leading N by N part of this array must contain the orthogonal matrix Q from the generalized Schur factorization. If FACT =.FALSE., Q need not be set on entry. LDQ { INTEGER. The leading dimension of the array Q as declared in the calling program. LDQ N. Z { DOUBLE PRECISION array of DIMENSION (LDZ,N). If FACT =.TRUE., then the leading N by N part of this array must contain the orthogonal matrix Z from the generalized Schur factorization. If FACT =.FALSE., Z need not be set on entry. LDZ { INTEGER. The leading dimension of the array Z as declared in the calling program. LDZ N. Arguments Out A { DOUBLE PRECISION array of DIMENSION (LDA,N). The leading N by N part of this array contains the generalized Schur factor A s of the matrix A. (A s is an upper quasitriangular matrix.) E { DOUBLE PRECISION array of DIMENSION (LDE,N). The leading N by N part of this array contains the generalized Schur factor E s of the matrix E. (E s is an upper triangular matrix.) X { DOUBLE PRECISION array of DIMENSION (LDX,N). If JOB = 'B' or 'X', then the leading N by N part of this array contains the solution matrix X of the equation. If JOB = 'S', X has not been referenced. SCALE { DOUBLE PRECISION. The scale factor set to avoid overow in X ( < SCALE ). Q { DOUBLE PRECISION array of DIMENSION (LDQ,N). The leading N by N part of this array contains the orthogonal matrix Q from the generalized Schur factorization. Z { DOUBLE PRECISION array of DIMENSION (LDZ,N). The leading N by N part of this array contains the orthogonal matrix Z from the generalized Schur factorization. SEP { DOUBLE PRECISION. An estimate for the separation of the Lyapunov operator. RCOND { DOUBLE PRECISION. An estimate for the reciprocal condition number of the Lyapunov operator. 3

14 Work Space IWORK { INTEGER array at least of DIMENSION (N2). If JOB = 'X', this array is not referenced. RWORK { DOUBLE PRECISION array at least of DIMENSION (LRWORK). On exit, if IERR =, RWORK() contains the optimal workspace. LRWORK { INTEGER. The dimension of the array RWORK. The following table contains the minimal work space requirements depending on the choice of JOB and FACT. JOB FACT LRWORK 'X'.TRUE. N 'X'.FALSE. 7N 'B', 'S'.TRUE. 2N 2 'B', 'S'.FALSE. max(2n 2 ; 7N) Note: For good performance, LRWORK must generally be larger. Tolerances None. Mode Parameters JOB { CHARACTER. Species if the solution is to be computed and if the separation (along with the reciprocal condition number) is to be estimated. JOB = 'X', JOB = 'S', JOB = 'B', DISCR { LOGICAL. (Compute the solution only); (Estimate the separation only); (Compute the solution and estimate the separation). Species which type of equation is to be solved. DISCR =.FALSE., (Continuous{time equation (23)); DISCR =.TRUE., (Discrete{time equation (24)). FACT { LOGICAL. Species whether the generalized real Schur factorization of the pencil A E is supplied on entry or not. FACT =.FALSE., FACT =.TRUE., TRANS { LOGICAL. (The generalized real Schur factorization is not supplied); (The generalized real Schur factorization is supplied). Species whether the transposed equation is to be solved or not. TRANS =.FALSE., (op(a)=a, op(e)=e); TRANS =.TRUE., (op(a)=a T, op(e)=e T ). UPPER { LOGICAL. Species whether the lower or the upper triangle of the array X is referenced. UPPER =.FALSE., UPPER =.TRUE., (Only the lower triangle is referenced); (Only the upper triangle is referenced). 4

15 Warning Indicator None. Error Indicator IERR { INTEGER. A..4 IERR = : Unless the routine detects an error (see next section), IERR contains on exit. Warnings and Errors detected by the Routine On entry, N <, or LDA < N, or LDE < N, or LDX < N, or LDQ < N, or LDZ < N, or JOB 62 f'b', 'b', 'S', 's', 'X', 'x'g. IERR = 2: IERR = 3: LRWORK too small. FACT =.TRUE. and the matrix contained in the upper Hessenberg part of the array A is not in upper quasitriangular form. IERR = 4: FACT =.FALSE. and the pencil A E cannot be reduced to generalized Schur form. LAPACK routine DGEGS has failed to converge. IERR = 5: DISCR =.TRUE. and the pencil A E has a pair of reciprocal eigenvalues. That is, i = = j for some i and j, where i and j are eigenvalues of A E. Hence, equation (24) is singular. IERR = 6: DISCR =.FALSE. and the pencil A E has a degenerate pair of eigenvalues. That is, i = j for some i and j, where i and j are eigenvalues of A E. Hence, equation (23) is singular. A.2 Hammarling's Method A.2. Purpose To compute the Cholesky factor U of the matrix X = op(u) T op(u), which is the solution of either the generalized c{stable continuous{time Lyapunov equation op(a) T X op(e) + op(e) T X op(a) = scale 2 op(b) T op(b) (25) or the generalized d{stable discrete{time Lyapunov equation op(a) T X op(a) op(e) T X op(e) = scale 2 op(b) T op(b) (26) without rst nding X and without the need to form the matrix op(b) T op(b). op(k) is either K or K T for K = A; B; E; U. A and E are N by N matrices, op(b) is an M by N matrix. The resulting matrix U is an N by N upper triangular matrix with non{negative entries on its main diagonal. scale is an output scale factor set to avoid overow in U. A.2.2 Specication SUBROUTINE DGLPHM( DISCR, FACT, TRANS, N, M, A, LDA, E, LDE, B, * LDB, SCALE, Q, LDQ, Z, LDZ, RWORK, LRWORK, * IERR ) 5

16 A.2.3 Arguments In N { INTEGER. Argument List The order of the matrix A. N. M { INTEGER. The number of rows in the matrix op(b). M. A { DOUBLE PRECISION array of DIMENSION (LDA,N). If FACT =.TRUE., then the leading N by N upper Hessenberg part of this array must contain the generalized Schur factor A s of the matrix A. A s must be an upper quasitriangular matrix. The elements below the upper Hessenberg part of the array A are not referenced. If FACT =.FALSE., then the leading N by N part of this array must contain the matrix A. Note: this array is overwritten if FACT =.FALSE.. LDA { INTEGER. The leading dimension of the array A as declared in the calling program. LDA N. E { DOUBLE PRECISION array of DIMENSION (LDE,N). If FACT =.TRUE., then the leading N by N upper triangular part of this array must contain the generalized Schur factor E s of the matrix E. The elements below the upper triangular part of the array E are not referenced. If FACT =.FALSE., then the leading N by N part of this array must contain the coecient matrix E of the equation. Note: this array is overwritten if FACT =.FALSE.. LDE { INTEGER. The leading dimension of the array E as declared in the calling program. LDE N. B { DOUBLE PRECISION array of DIMENSION (LDB,N). If TRANS =.TRUE., the leading N by M part of this array must contain the matrix B and N MAX(M,N). If TRANS =.FALSE., the leading M by N part of this array must contain the matrix B and N N. Note: this array is overwritten. LDB { INTEGER. The leading dimension of the array B as declared in the calling program. If TRANS =.TRUE., then LDB N. If TRANS =.FALSE., then LDB MAX(M,N). Q { DOUBLE PRECISION array of DIMENSION (LDQ,N). If FACT =.TRUE., then the leading N by N part of this array must contain the orthogonal matrix Q from the generalized Schur factorization. If FACT =.FALSE., Q need not be set on entry. LDQ { INTEGER. The leading dimension of the array Q as declared in the calling program. LDQ N. 6

17 Z { DOUBLE PRECISION array of DIMENSION (LDZ,N). If FACT =.TRUE., then the leading N by N part of this array must contain the orthogonal matrix Z from the generalized Schur factorization. If FACT =.FALSE., Z need not be set on entry. LDZ { INTEGER. The leading dimension of the array Z as declared in the calling program. LDZ N. Arguments Out A { DOUBLE PRECISION array of DIMENSION (LDA,N). The leading N by N part of this array contains the generalized Schur factor A s of the matrix A. (A s is an upper quasitriangular matrix.) E { DOUBLE PRECISION array of DIMENSION (LDE,N). The leading N by N part of this array contains the generalized Schur factor E s of the matrix E. (E s is an upper triangular matrix.) B { DOUBLE PRECISION array of DIMENSION (LDB,N). The leading N by N part of this array contains the Cholesky factor U of the solution matrix X of the problem. SCALE { DOUBLE PRECISION. The scale factor set to avoid overow in U ( < SCALE ). Q { DOUBLE PRECISION array of DIMENSION (LDQ,N). The leading N by N part of this array contains the orthogonal matrix Q from the generalized Schur factorization. Z { DOUBLE PRECISION array of DIMENSION (LDZ,N). The leading N by N part of this array contains the orthogonal matrix Z from the generalized Schur factorization. Work Space RWORK { DOUBLE PRECISION array at least of DIMENSION (LRWORK). On exit, if IERR =, RWORK() contains the optimal workspace. LRWORK { INTEGER. The dimension of the array RWORK. If FACT =.TRUE., then LRWORK MAX(6N-6,). If FACT =.FALSE., then LRWORK MAX(7N,). Note: For good performance, LRWORK must generally be larger. Tolerances None. Mode Parameters DISCR { LOGICAL. Species which type of equation is to be solved. DISCR =.FALSE., (Continuous{time equation (25)); DISCR =.TRUE., (Discrete{time equation (26)). 7

18 FACT { LOGICAL. Species whether the generalized real Schur factorization of the pencil A E is supplied on entry or not. FACT =.FALSE., FACT =.TRUE., TRANS { LOGICAL. (The generalized real Schur factorization is not supplied); (The generalized real Schur factorization is supplied). Species whether the transposed equation is to be solved or not. TRANS =.FALSE., TRANS =.TRUE., Warning Indicator None. Error Indicator IERR { INTEGER. A.2.4 IERR = : (op(k)=k, K = A; B; E; U); (op(k)=k T, K = A; B; E; U). Unless the routine detects an error (see next section), IERR contains on exit. Warnings and Errors detected by the Routine On entry, N <, or M <, or LDA < N, or LDE < N, or LDB < N, or LDQ < N, or LDZ < N, or (TRANS =.FALSE. and LDB < M). IERR = 2: IERR = 3: LRWORK too small. FACT =.TRUE. and the matrix contained in the upper Hessenberg part of the array A is not in upper quasitriangular form. IERR = 4: FACT =.FALSE. and the pencil A E cannot be reduced to generalized Schur form. LAPACK routine DGEGS has failed to converge. IERR = 5: FACT =.TRUE. and there is a 2 by 2 block on the main diagonal of the pencil A s E s with real eigenvalues. IERR = 6: IERR = 7: IERR = 8: DISCR =.FALSE. and the pencil A E is not c{stable. DISCR =.TRUE. and the pencil A E is not d{stable. DISCR =.TRUE. and the LAPACK routine DSYEVX has failed to converge during the solution of the reduced equation. This error is unlikely to occur. 8

19 B Example Programs Two sample programs are enclosed to demonstrate the usage of the routines DGLP and DGLPHM. B. Bartels{Stewart Method Example To nd the solution matrix X, the separation, and the reciprocal condition number of the equation where A T XE + E T XA = Y; A Program Text 3: : : : 3: : : : 2: A ; E : 3: : 3: 2: : : : : A ; and Y 64: 73: 28: 73: 7: 25: 28: 25: 8: * DGLP EXAMPLE PROGRAM TEXT * *.. Parameters.. INTEGER NIN, NOUT PARAMETER (NIN=5, NOUT=6) INTEGER NMAX PARAMETER (NMAX=2) INTEGER LDA, LDE, LDQ, LDX, LDZ PARAMETER (LDA=NMAX, LDE=NMAX, LDQ=NMAX, LDX=NMAX, + LDZ=NMAX) INTEGER LIWORK, LRWORK PARAMETER (LIWORK=NMAX**2, LRWORK=MAX(2*NMAX**2,7*NMAX)) *.. Local Scalars.. CHARACTER JOB DOUBLE PRECISION RCOND, SCALE, SEP INTEGER I, IERR, J, N LOGICAL DISCR, FACT, TRANS, UPPER *.. Local Arrays.. INTEGER IWORK(LIWORK) DOUBLE PRECISION A(LDA,NMAX), E(LDE,NMAX), Q(LDQ,NMAX), + RWORK(LRWORK), X(LDX,NMAX), Z(LDZ,NMAX) *.. External Subroutines.. EXTERNAL DGLP *.. Executable Statements.. * WRITE (NOUT,FMT=99999) * Skip the heading in the data file and read the data. READ (NIN,FMT='()') READ (NIN,FMT=*) N, JOB, DISCR, FACT, TRANS, UPPER IF (N.LE..OR. N.GT.NMAX) THEN WRITE (NOUT,FMT=99993) N ELSE READ (NIN,FMT=*) ((A(I,J),J=,N),I=,N) READ (NIN,FMT=*) ((E(I,J),J=,N),I=,N) IF (FACT) THEN READ (NIN,FMT=*) ((Q(I,J),J=,N),I=,N) READ (NIN,FMT=*) ((Z(I,J),J=,N),I=,N) END IF READ (NIN,FMT=*) ((X(I,J),J=,N),I=,N) * Find the solution matrix X and the scalars RCOND and SEP. A : 9

20 CALL DGLP(JOB,DISCR,FACT,TRANS,N,A,LDA,E,LDE,UPPER,X,LDX,SCALE, + Q,LDQ,Z,LDZ,IWORK,RWORK,LRWORK,SEP,RCOND,IERR) * IF (IERR.NE.) THEN WRITE (NOUT,FMT=99998) IERR ELSE IF (JOB.EQ.'B'.OR.JOB.EQ.'S') THEN WRITE (NOUT,FMT=99997) SEP WRITE (NOUT,FMT=99996) RCOND END IF IF (JOB.EQ.'B'.OR.JOB.EQ.'X') THEN WRITE (NOUT,FMT=99995) SCALE DO 2 I =, N WRITE (NOUT,FMT=99994) (X(I,J),J=,N) 2 CONTINUE END IF END IF END IF STOP * FORMAT (' DGLP EXAMPLE PROGRAM RESULTS',/X) FORMAT (' IERR on exit from DGLP = ',I2) FORMAT (' SEP = ',F8.4) FORMAT (' RCOND = ',F8.4) FORMAT (' SCALE = ',F8.4,//' The solution matrix X is ') FORMAT (2(X,F8.4)) FORMAT (/' N is out of range.',/' N = ',I5) END Program Data DGLP EXAMPLE PROGRAM DATA 3 B F F F T Program Results DGLP EXAMPLE PROGRAM RESULTS SEP =.2867 RCOND =.55 SCALE =. The solution matrix X is

21 B.2 Hammarling's Method Example To nd the Cholesky factor U of the solution X = U T U of the equation where A T XE + E T XA = B T B; A Program Text : 3: 4: : 5: 2: 4: 4: : A ; E 2: : 3: 2: : : 4: 5: : A ; and B = 2: : 7: : * DGLPHM EXAMPLE PROGRAM TEXT * *.. Parameters.. INTEGER NIN, NOUT PARAMETER (NIN=5, NOUT=6) INTEGER NMAX PARAMETER (NMAX=2) INTEGER LDA, LDB, LDE, LDQ, LDZ PARAMETER (LDA=NMAX, LDB=NMAX, LDE=NMAX, LDQ=NMAX, + LDZ=NMAX) INTEGER LRWORK PARAMETER (LRWORK=MAX(7*NMAX,6*NMAX-6,)) *.. Local Scalars.. DOUBLE PRECISION SCALE INTEGER I, IERR, J, N, M LOGICAL DISCR, FACT, TRANS *.. Local Arrays.. DOUBLE PRECISION A(LDA,NMAX), B(LDB,NMAX), E(LDE,NMAX), + Q(LDQ,NMAX), RWORK(LRWORK), Z(LDZ,NMAX) *.. External Subroutines.. EXTERNAL DGLPHM *.. Executable Statements.. * WRITE (NOUT,FMT=99999) * Skip the heading in the data file and read the data. READ (NIN,FMT='()') READ (NIN,FMT=*) N, M, DISCR, FACT, TRANS IF (N.LT..OR. N.GT.NMAX) THEN WRITE (NOUT,FMT=99995) N ELSEIF (M.LT..OR. M.GT.NMAX) THEN WRITE (NOUT,FMT=99994) M ELSE READ (NIN,FMT=*) ((A(I,J),J=,N),I=,N) READ (NIN,FMT=*) ((E(I,J),J=,N),I=,N) IF (FACT) THEN READ (NIN,FMT=*) ((Q(I,J),J=,N),I=,N) READ (NIN,FMT=*) ((Z(I,J),J=,N),I=,N) END IF IF (TRANS) THEN READ (NIN,FMT=*) ((B(I,J),J=,M),I=,N) ELSE READ (NIN,FMT=*) ((B(I,J),J=,N),I=,M) END IF * Find the Cholesky factor U of the solution matrix. CALL DGLPHM(DISCR,FACT,TRANS,N,M,A,LDA,E,LDE,B,LDB,SCALE,Q,LDQ, 2

22 + Z,LDZ,RWORK,LRWORK,IERR) * IF (IERR.NE.) THEN WRITE (NOUT,FMT=99998) IERR ELSE WRITE (NOUT,FMT=99997) SCALE DO 2 I =, N WRITE (NOUT,FMT=99996) (B(I,J),J=,N) 2 CONTINUE END IF END IF STOP * FORMAT (' DGLPHM EXAMPLE PROGRAM RESULTS',/X) FORMAT (' IERR on exit from DGLPHM = ',I2) FORMAT (' SCALE = ',F8.4,//' The Cholesky factor U of the solution + matrix is') FORMAT (2(X,F8.4)) FORMAT (/' N is out of range.',/' N = ',I5) FORMAT (/' M is out of range.',/' M = ',I5) END Program Data DGLPHM EXAMPLE PROGRAM DATA 3 F F F Program Results DGLPHM EXAMPLE PROGRAM RESULTS SCALE =. The Cholesky factor U of the solution matrix is

23 References [] E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorenson, LAPACK User's Guide, Philadelphia, PA, 992. [2] R.H. Bartels, G.W. Stewart, Solution of the Equation AX + XB = C, Comm. A.C.M., 5:82{826, 972. [3] K.{W.E. Chu, The solution of the matrix equation AXB CXD = Y and (Y A DZ; Y C BZ) = (E; F ), Linear Algebra Appl., 93:93{5, 987. [4] J.J. Dongarra, J.R. Bunch, C.B. Moler, G.W. Stewart, LINPACK User's Guide, Philadelphia, PA, 979. [5] B.S. Garbow, J.M. Boyle, J.J. Dongarra, C.B. Moler, Matrix Eigensystem Routines { EISPACK Guide Extension, Lecture Notes in Computer Science, Vol.5, Springer{Verlag, New York, 977. [6] J.D. Gardiner, A.J. Laub, J.J. Amato, C.B. Moler, Solution of the Sylvester Matrix Equation AXB T + CXD T = E, A.C.M. Trans. Math. Soft., 8:223{23, 992. [7] J.D. Gardiner, M.R. Wette, A.J. Laub, J.J. Amato, C.B. Moler, A FORTRAN{77 Software Package for Solving the Sylvester Matrix Equation AXB T + CXD T = E, A.C.M. Trans. Math. Soft., 8:232{238, 992. [8] G.H. Golub, C.F. Van Loan, Matrix Computations, John Hopkins University Press, Baltimore, Second Edition, 989. [9] W.W. Hager, Condition Estimates, SIAM J. Sci. Stat. Comput., 5:3{36, 984. [] S.J. Hammarling, Numerical Solution of the Stable, Non{negative Denite Lyapunov Equation, IMA J. Num. Anal., 2:33{323, 982. [] N.J. Higham, FORTRAN Codes for Estimating the One{Norm of a Real or Complex Matrix, with Applications to Condition Estimation, A.C.M. Trans. Math. Soft., 4:38-396, 988. [2] B. Kagstrom, L. Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Trans. Aut. Control, 34:745-75, 989. [3] P. Lancaster, M. Tismenetsky, The Theory of Matrices, Academic Press, Orlando, 2nd edition, 985. [4] J. Snyders, M. Zakai, On Non{negative Solutions of the Equation AD + DA T = C, SIAM J. Appl. Math., 8:74-74, 97. [5] Working Group on Software (WGS), Implementation and Documentation Standards, WGS{Report 9{, Eindhoven/Oxford,